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Texas hold ’em – Starting hands heads up

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For any given starting hand, there are 50 × 49 ÷ 2 = 1,225 hands that an opponent can have before the flop. (After the flop, the number of possible hands an opponent can have is reduced by the three community cards revealed on the flop to 47 × 46 ÷ 2 = 1,081 hands.) Therefore, there are

(52/2)(50/2) -: 2 = 812.175

possible head-to-head match ups in Hold ’em. (The number of total number of match ups is divided by the two ways that two hands can be distributed between two players to give the number of unique match ups.) However, since there are only 169 distinct starting hands, there are 169 × 1,225 = 207,025 distinct head-to-head match ups.

It is useful and interesting to know how two starting hands compete against each other heads up before the flop. In other words, we assume that neither hand will fold, and we will see a showdown. This situation occurs quite often in no limit and tournament play. Also, studying these odds helps to demonstrate the concept of hand domination, which is important in all community card games.

This problem is considerably more complicated than determining the frequency of dealt hands. To see why, note that given both hands, there are 48 remaining unseen cards. Out of these 48 cards, we can choose any 5 to make a board. Thus, there are

(48/5) = 1.712.304

possible boards that may fall. In addition to determining the precise number of boards that give a win to each player, we also must take into account boards which split the pot, and split the number of these boards between the players.

The problem is trivial for computers to solve by brute force search; there are many software programs available that will compute the odds in seconds. A somewhat less trivial exercise is an exhaustive analysis of all of the head-to-head match ups in Texas Hold ’em, which requires evaluating each possible board for each distinct head-to-head match up, or 1,712,304 × 207,025 = 354,489,735,600 (≈354 billion) results.

When comparing two starting hands, the head-to-head probability describes the likelihood of one hand beating the other after all of the cards have come out. Head-to-head probabilities vary slightly for each particular distinct starting hand matchup, but the approximate average probabilities, as given by Dan Harrington in Harrington on Hold’em [p.125], are summarized in the following table.

Favorite-to-underdog matchup Probability Odds for
Pair vs. 2 undercards 0.83 4.9 : 1
Pair vs. lower pair 0.82 4.5 : 1
Pair vs. 1 overcard, 1 undercard 0.71 2.5 : 1
2 overcards vs. 2 undercards 0.63 1.7 : 1
Pair vs. 2 overcards 0.55 1.2 : 1

These odds are general approximations only derived from averaging all of the hand matchups in each category. The actual head-to-head probabilities for any two starting hands vary depending on a number of factors, including:

• Suited or unsuited starting hands;
• Shared suits between starting hands;
• Connectedness of non-pair starting hands;
• Proximity of card ranks between the starting hands (lowering straight potential);
• Proximity of card ranks toward A or 2 (lowering straight potential);
• Possibility of split pot.

For example, A♠ A♣ vs. K♠ Q♣ is 87.65% to win (0.49% to split), but A♠ A♣ vs. 7♦ 6♦ is 76.81% to win (0.32% to split).

The mathematics for computing all of the possible matchups is quite complex. However, a computer program can perform a brute force evaluation of the 1,712,304 possible boards for any given pair of starting hands in seconds.

Notes

1. ^ a b By removing reflection and applying aggressive search tree pruning, it is possible to reduce the number of unique head-to-head hand combinations from 207,205 to less than 50,000. Reflection eliminates redundant calculations by observing that given hands h1 and h2, if w1 is the probability of h1 beating h2 in a showdown and s is the probability of h1 splitting the pot with h2, then the probability w2 of h2 beating h1 is w2 = 1 − (s + w1), thus eliminating the need to evaluate h2 against h1. Pruning is possible, for example, by observing that Q♥ J♥ has the same chance of winning against both 8♦ 7♣ and 8♦ 7♠ (but not the same probability as against 8♥ 7♣ because sharing the heart affects the flush possibilities for each hand.)